Procesado de Señales e Imágenes Médicas

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2024-08-12

Introduction to Wavelet Transform

Introduction

  • It’s a mathematical tool for signal decomposition, like Fourier’s Transform.
  • Just as the Fourier transform decomposes a signal into a series of sine and cosine functions, the wavelet transform does so using a set of functions known as wavelets.
  • Wavelets are functions generated by scaling and shifting a base function known as the mother wavelet.

Introduction

Introduction

  • Morlet: Popular for time-frequency analysis in EEG and ECG.
  • Mexican Hat (Ricker): Often used in spike detection in neural signals.
  • Haar: Useful in quick decomposition of signals and feature extraction.
  • Daubechies: Frequently used in ECG signal denoising and compression.
  • Symlet: Another option for signal processing and feature extraction in EEG.
  • Coiflet: Useful for denoising and baseline correction in biomedical signals.

Introduction

  • Have a mean of zero (to capture details in the signal).
  • Be square integrable (finite energy).
  • Satisfy the admissibility condition on its Fourier transform.
  • Be oscillatory to capture frequency information.
  • (Optionally) have compact support for efficient computation and localization.

Zero Mean (Admissibility Condition)

The function must have an average value of zero. Mathematically, this is expressed as:

\[\int_{-\infty}^{\infty} \psi(t) \, dt = 0\]

This condition ensures that the wavelet can detect changes or “details” in the signal rather than its average or constant components.

Square Integrability

The function \(\psi(t)\) must be square integrable, meaning it has finite energy:

\[\int_{-\infty}^{\infty} |\psi(t)|^2 \, dt < \infty\]

This requirement ensures that the wavelet’s energy is finite, making it possible to localize the function in both time and frequency domains. Functions that satisfy this belong to the \(L^2(\rm I\!R)\) space, which is the space of all functions with finite energy.

Admissibility Constant

The wavelet’s Fourier transform, \(\hat{\psi}(\omega)\), should satisfy the admissibility condition:

\[C_\psi = \int_{-\infty}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{|\omega|} \, d\omega < \infty\]

where \(\hat{\psi}(\omega)\) is the Fourier transform of \(\psi(t)\), and \(\omega\) represents angular frequency. This condition implies that \(\hat{\psi}(\omega)\) must approach zero as \(\omega \rightarrow 0\) meaning the wavelet has no component at zero frequency (or DC component). This condition is crucial for ensuring that the wavelet transform is invertible.

Oscillatory Nature

A mother wavelet should generally be oscillatory or “wavelike” (hence the term “wavelet”). This oscillatory behavior allows the wavelet to capture variations in the signal. For example, wavelets like the Morlet wavelet resemble decaying sinusoids. This oscillatory nature helps the wavelet capture both high-frequency and low-frequency components effectively.

Compact Support

Although not strictly necessary, compact support is often a desirable property. Compact support means that the function is non-zero only over a finite interval, making it well-localized in time. This allows for efficient computation and good localization in the time domain. For example, the Haar wavelet has compact support, while others, like the Morlet wavelet, do not have strict compact support but still decay rapidly.

A wavelet transformation

(0.0, 1.0)

A wavelet transformation

Mathematical Expressions

The continuous wavelet transform of a signal \(f(t)\) is defined as:

\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]

where:

  • \(f(t)\) is the input signal,
  • \(\psi\) is the mother wavelet,
  • \(a\) is the scale parameter (controls the width of the wavelet),
  • \(b\) is the translation parameter (controls the position of the wavelet),
  • \(\psi^*\) denotes the complex conjugate of the mother wavelet.